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Tricky inequality

Source: JBMO 2023 Problem 2

June 26, 2023
Inequalitythree variable inequalityinequalities

Problem Statement

Prove that for all non-negative real numbers x,y,zx,y,z, not all equal to 00, the following inequality holds
2x2−x+y+zx+y2+z2+2y2+x−y+zx2+y+z2+2z2+x+y−zx2+y2+z≥3.\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.
Determine all the triples (x,y,z)(x,y,z) for which the equality holds.
Milan Mitreski, Serbia
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